Let’s remind ourselves of some important facts about angles:

• How many degrees are there in a complete turn?
• How many degrees are there in a straight line?

Example

There are some important angle relationships when a transversal line crosses a pair of parallel lines. In this case, can we identify and specify what is important about, the following types of angle:

• Vertically opposite;
• Corresponding;
• Alternate.

Example

Exercise

Let’s complete exercise 9A and 9B on pages 128 and 129 of the textbook:

The answers are below:

(1.) (a) 115º, (b) 45º, (c) 47º, (d) 50º, (2.) (a) 33º, (b) 97º, (c) 52º, (d) 40º, (3.) r, (4.) (a) 80º, (b) 143º, (c) 131º, (d) 72º

(1.) (a) 77º, (b) 103º, (c) 48º, (d) 92º, 92º, (2.) (a) x=120º, y=60º, z=80º, (b) p=103º, q=77º, r=103º, s=77º, (c) l=m=n=78º, (3.) (a) 72º, (b) 108º, (4.) a = 123º, b=83º, c=140º, d=97º, (5.) a=36º, b=72º, (6.) x=30º, y=120º, z=60º.

Angles in triangles and quadrilaterals

What is the sum of the angles in any triangle?

What about a quadrilateral (a 4-sided shape)?

Examples

Exercise

Let’s complete exercise 9C on pages 130 and 131 of the textbook:

The answers are below:

Angles in polygons

What is a polygon?

What is an interior angle of a polygon? What is an exterior angle of a polygon?

How can we find out the total number of degrees in a polygon (hint: triangles are our friends)

Example

Exercise

Let’s complete exercise 9D on page 132 of the textbook:

The answers are below:

Exterior angles

Consider a regular polygon. Look at all of its exterior angles. What must their sum be?

So if you want to know the size of an exterior angle in a regular polygon, how can you calculate it?

How could you use this to help you find the size of the interior angles in a regular polygon?

Example

Exercise

Let’s complete exercise 9E on page 134 of the textbook:

The answers are below:

Angles in circles

There are lots of special relationships related to angles in circles. First though, let’s make sure we know the names of some of the important parts of a circle:

The first thing we want to notice about angles in a circle is the relationship between the angles that a chord subtends at two different points on the circumference of the circle.

The second thing we want to notice about angles in a circle is the relationship between the angle that a chord subtends on the circumference of the circle and the angle that the same chord subtends at the centre of the circle.

Exercise

Let’s complete exercise 9F on pages 135 to 136 of the textbook:

The answers are below:

There are three more properties of the angles in a circle that we need to know

• What is the angle that the diameter subtends on the circumference of the circle;
• What is the sum of the opposite interior angles in a cyclic quadrilateral?
• What is the angle made between a tangent of a circle and a radius of a circle?

Exercise

Let’s complete exercise 9G on pages 137 to 139 of the textbook:

The answers are below: